A 2026 weak-curvature F-theory QCD-axion ensemble constraint

Given: The paper's finite model ensembles, its selected weak-curvature Kähler points and QCD-axion mass approximation, and its source-specific interpretation of the SN1987A neutrino-burst constraint.

Table I reports 14,627 Tree models with $35\le N\le \mathrm{2,591}$, 16,556 Skeleton models with $10\le N\le \mathrm{15,415}$, and 54 K2 models with $\mathrm{2,560}\le N\le \mathrm{181,200}$. The source does not give precise per-bin counts. Its high-$N$ bins therefore contain no Tree models. Fallon et al., Table I and Results

The source adopts the approximate $15\mathrm{m}\mathrm{e}\mathrm{V}$ QCD-axion mass limit from its axion-nucleon emissivity and Raffelt-criterion analysis of the SN1987A neutrino burst. The criterion is based on one-dimensional supernova simulations; it is an astrophysical inference, not a measured F-theory quantity. The paper does not independently audit the underlying supernova observations or simulations. Fallon et al., SN1987A and the QCD axion

For each constructed model, the authors select a weak-curvature Kähler point using their stated gauge-coupling and supersymmetry-scale conditions, then estimate $m_{\mathrm{Q}\mathrm{C}\mathrm{D}}$ with their effective-field-theory procedure and a selected subset of instanton scales. Weak curvature is used to argue that ${\alpha }^{\mathrm{\prime }}$ corrections are plausibly negligible; this is a source approximation, not a proved error bound. Fallon et al., Methodology

The Results section and Fig. 3 report that in every populated bin with $N\ge \mathrm{8,791}$, at least $95\mathrm{\%}$ of the sampled models lie above the adopted $15\mathrm{m}\mathrm{e}\mathrm{V}$ threshold. The paper gives no precise model count for each such bin. This is the authors' finite-sample report, not a probability statement for a larger population. Fallon et al., Results and Fig. 3

Compact Kähler and rational Hodge polarization conventions fixes the compact Kähler convention for the source's base.

Compact Kähler Hodge decomposition from elliptic harmonic forms identifies $H^{1,1}(B_3)$ with the bidegree-$(1,1)$ Hodge summand and the Dolbeault group $H^1(B_3,{\mathrm{\Omega }}_{B_3}^1)$; it supplies finite-dimensionality under compact-Kähler and full-AC hypotheses.

Type-IIB axiodilaton and elliptic-fiber dictionary adopts the geometric elliptic-fibration/F-theory dictionary. Its current statement supplies the context used here; this item does not use it as a proof of the reported ensemble frequency.